What is a normal field?

In mathematics, a normal field is a field extension that is both algebraic and separable. Specifically, a field extension ( K/F ) is normal if every irreducible polynomial in ( F[x] ) that has at least one root in ( K ) splits completely into linear factors in ( K ). This property ensures that all roots of these polynomials are contained within the extension, making normal fields important in the study of algebraic structures and Galois theory.